Can we take the derivative of functionals on the space of smooth scalar fields of a riemannian manifold?

Question

Is it possible to take the derivative of a functional $ \mathcal{L}:\mathcal{F} \rightarrow \mathbb{R} $, where $\mathcal{F} = \{f:\mathcal{M} \rightarrow \mathbb{R}|f \: is \: smooth\} $. We know the riemannian manifold $\mathcal{M}$. We would like to derive the derivative of some simple functionals(like the evaluation functional) that depend on the riemanian metric tensor. Also I would appreciate some references for further reading.